For the purpose of diagnosis, the TX is set to transmit symbol \(x=1\)  throughout the diagnosis.
Besides, to eliminate the received signal contributed by the direct channel \(h_\mathrm {D}\) , we first set the IRS to the absorbing mode (i.e., \(\beta _n = 0\)  for \(n=1, \cdots , N\) ) before the diagnosis and conduct channel estimation to obtain an estimator of \(h_\mathrm {D}\)  as \(\hat{h}_\mathrm {D}\)  [1]}, and then subtract it from \(y^{\prime }\) .
Therefore, the transmission model for diagnosis can be rewritten as
\(\begin{aligned}y&= y^{\prime }|_{x=1} - \hat{h}_\mathrm {D} \\&= \mathbf {h}_\mathrm {RX}^\mathrm {T} \bigg (\Gamma  \mathrm {diag}(\mathbf {m})\bigg ) \mathbf {h}_\mathrm {TX} + \underbrace{h_\mathrm {D} - \hat{h}_\mathrm {D} + w^{\prime }}_{=w} \\&= \mathbf {h}_\mathrm {RX}^\mathrm {T} \bigg (\mathrm {diag}(\mathbf {f}) \mathrm {diag}(\mathbf {m}) \bigg ) \mathbf {h}_\mathrm {TX} + w \\&= \sum _{n=1}^{N} \left[\mathbf {h}_\mathrm {RX} \circ \mathbf {f} \circ \mathbf {m} \circ \mathbf {h}_\mathrm {TX}\right]_n + w,\end{aligned}\) 
